A combined (fixed-J) two-dimensional master-equation/semi-classical transition state theory/variational Rice-Ramsperger-Kassel-Marcus approach has been used to compute reaction rate coefficients of •OH with CH3OH over a wide range of temperatures (10-2500 K) and pressures (10-1-104 Torr) based on a potential energy surface that has been constructed using a modification of the high accuracy extrapolated ab initio thermochemistry (HEAT) protocol. The calculated results show that the title reaction is nearly pressure-independent when T > 250 K but depends strongly on pressure at lower temperatures. In addition, the preferred mechanism and rate constants are found to be very sensitive to temperature. The reaction pathway CH3OH + •OH → CH3O• + H2O proceeds exclusively through tunneling at exceedingly low temperatures (T ≤ 50 K), typical of those established in interstellar environments. In this regime, the rate constant is found to increase with decreasing temperature, which agrees with low-temperature experimental results. The thermodynamically favored reaction pathway CH3OH + •OH → •CH2OH + H2O becomes dominant at higher temperatures (T ≥ 200 K), such as those found in Earth's atmosphere as well as combustion environments. By adjusting the ab initio barrier heights slightly, experimental rate constants from 200 to 1250 K can be satisfactorily reproduced.